# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- # **Key Concepts** This exercise focuses on the following concepts: * **Intersection of Lines and Hyperbolas:** Calculating chord length using the distance formula and substitution. * **Condition of Tangency:** Applying the condition $c^2 = a^2m^2 - b^2$ for a line $y = mx + c$ to be tangent to a hyperbola. * **Tangent Equations:** Determining equations of tangents based on slope (parallel or perpendicular to given lines). * **Point Form Equations:** Using the $T=0$ method to find tangents and normals at specific points $(x_1, y_1)$. * **Shifted Conics:** Handling hyperbolas with translated centers $(h, k)$. * **Geometric Properties:** Understanding the reflective property of the hyperbola involving focal radii. * **Point of Contact:** Identifying the specific coordinate where a tangent touches the curve. --- # **Important Formulas** Below are the key formulas used in this exercise: | Description | Formula | | :--- | :--- | | Standard Hyperbola | $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ | | Condition of Tangency | $c^2 = a^2m^2 - b^2$ | | Equation of Tangent (Slope $m$) | $y = mx \pm \sqrt{a^2m^2 - b^2}$ | | Equation of Tangent at $(x_1, y_1)$ | $\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1$ | | Equation of Normal at $(x_1, y_1)$ | $\frac{a^2x}{x_1} + \frac{b^2y}{y_1} = a^2 + b^2$ | | Point of Contact | $(\pm \frac{a^2m}{c}, \pm \frac{b^2}{c})$ | --- # **Summary** This exercise focuses on the linear properties of hyperbolas, specifically tangents and normals. The primary strategy involves converting general second-degree equations into standard or shifted forms to identify $a^2$ and $b^2$. For slope-related problems, the condition $c^2 = a^2m^2 - b^2$ is the most efficient tool. For problems involving specific points, the substitution method ($x^2 \to xx_1$, $y^2 \to yy_1$, etc.) provides the tangent equation directly. A key learning is the geometric property that the tangent at any point bisects the angle between the focal radii.